Longitudinal Waves
madhatter,
Not to stumble off into quaternions and vector calculus, but are you using the Heaviside-Hertz duplex form of "Maxwell's Equations"[1] in your description? Heaviside vehemently opposes even the thought of a longitudinal wave in this formulation of Maxwell's theory[2]. That said, ET Whittaker[4] and Helmholtz[5], aside from Tesla, thought that longitudinal electric waves could exist and worked these problems out mathematically, something Tesla never did or at least never published. Whittaker starts to delve into an alternative theory for gravity while also developing a theory for longitudinal electric waves utilizing Maxwell's "discarded" potentials. Interestingly, Helmholtz also makes use of the potentials in his work, but Heaviside finds it lacking[2].
While I'm not qualified to give an opinion, I don't think the vector duplex form of Maxwell's equations can accurately describe a longitudinal electric wave. This being the fundamental issue people face when working on this topic in general. It would seem that the "potentials", while nearly useless for describing transverse wave mechanics[3], may indeed be needed for the consideration of a longitudinal wave.
T.E. Bearden gives a great overview of the situation in his paper "Maxwell's Lost Unified Field Theory of Electromagnetics & Gravitation". Where he has painstakingly given his theory along with a plethora of useful references and notes for those unacquainted with all the details. Bearden seems to believe that quaternions and Maxwell's potentials are the only way you can fully describe a longitudinal electric wave.
More to be added later...
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Notes & References:
[1] Oliver Heaviside and Heinrich Hertz both independently discarded Maxwell's "potentials" and through such developed the "duplex form" of what we currently call "Maxwell's Equations" utilizing E, H, D & B.
O. Heaviside E.M.T. Vol. 1, pgs. iii-iv
The term "Maxwell's Equations" becomes a bit misleading as the modern form is actually a truncated subset of its original form, not withstanding the conversion from quaternions to vectors. The reasoning behind the continued use of the phrase "Maxwell's Equations" probably stems from Heaviside's lengthy commentary as given in the preface of his first volume on electromagnetic theory (notably, the text in red):
O. Heaviside E.M.T. Vol. 1, pgs. vi-vii
[2] "On Compressional Electric or Magnetic Waves", O. Heaviside E.M.T. Vol. 2, Appendix D pgs. 493-506
[3] "On the Metaphysical Nature of the Propagation of the Potentials" O. Heaviside,Phil. Mag. 1889, Series 5, Vol. 27, No. 164, pgs 47-50
[4a] "The Partial Differential Equations of Mathematical Physics", E.T. Whittaker Mathematical Physics Vol. 57, 1903, pgs. 333-355
[4b] "On An Expression of the Electromagnetic Field Due to Electrons by Means of Two Scalar Potential Functions", E.T. Whittaker London Mathematical Society, 1904, Series 2, Vol. 1, pgs 367-372
[5] Helmholtz's paper is hard to come by, however, in reference [2] Heaviside brings it up and gives some commentary.
Regards,
Garrett M
madhatter,
Not to stumble off into quaternions and vector calculus, but are you using the Heaviside-Hertz duplex form of "Maxwell's Equations"[1] in your description? Heaviside vehemently opposes even the thought of a longitudinal wave in this formulation of Maxwell's theory[2]. That said, ET Whittaker[4] and Helmholtz[5], aside from Tesla, thought that longitudinal electric waves could exist and worked these problems out mathematically, something Tesla never did or at least never published. Whittaker starts to delve into an alternative theory for gravity while also developing a theory for longitudinal electric waves utilizing Maxwell's "discarded" potentials. Interestingly, Helmholtz also makes use of the potentials in his work, but Heaviside finds it lacking[2].
While I'm not qualified to give an opinion, I don't think the vector duplex form of Maxwell's equations can accurately describe a longitudinal electric wave. This being the fundamental issue people face when working on this topic in general. It would seem that the "potentials", while nearly useless for describing transverse wave mechanics[3], may indeed be needed for the consideration of a longitudinal wave.
T.E. Bearden gives a great overview of the situation in his paper "Maxwell's Lost Unified Field Theory of Electromagnetics & Gravitation". Where he has painstakingly given his theory along with a plethora of useful references and notes for those unacquainted with all the details. Bearden seems to believe that quaternions and Maxwell's potentials are the only way you can fully describe a longitudinal electric wave.
More to be added later...
-----------------------------------------------------------------
Notes & References:
[1] Oliver Heaviside and Heinrich Hertz both independently discarded Maxwell's "potentials" and through such developed the "duplex form" of what we currently call "Maxwell's Equations" utilizing E, H, D & B.
"The second chapter, pp. 20 to 131, consists of an outline scheme of the fundamentals of electromagnetic theory from the Faraday-Maxwell point of view, with some small modifications and extensions upon Maxwell's equations. It is done in terms of my rational units, which furnish the only way of carrying out the idea of lines and tubes of force in a consistent and ineligible manner. It is also done mainly in terms of vectors, for the sufficient reason that vectors are the main subject of investigation. It is also done in the duplex form I introduced in 1885, whereby the electric and magnetic sides of electromagnetism are symmetrically exhibited and connected, whilst the 'forces' and 'fluxes' are the objects of immediate attention, instead of the potential functions which are such powerful aids to obscuring and complicating the subject, and hiding from view useful and sometimes important relations."
The term "Maxwell's Equations" becomes a bit misleading as the modern form is actually a truncated subset of its original form, not withstanding the conversion from quaternions to vectors. The reasoning behind the continued use of the phrase "Maxwell's Equations" probably stems from Heaviside's lengthy commentary as given in the preface of his first volume on electromagnetic theory (notably, the text in red):
"It is essentially Maxwell's theory, but there are some differences. Some are changes of form only; for instance, the rationalization effected by changing the units, and the substitution of the second circuital law for Maxwell's equation of electromotive force involving the potentials, etc. But there is one change in particular which raises a fresh question. What is Maxwell's theory? What should we agree to understand by Maxwell's theory?
The first approximation to the answer is to say, There is Maxwell's book as he wrote it; there is his text, and there are his equations: together they make his theory. But when we come to examine it closely, we find that this answer is unsatisfactory.
To begin with, it is sufficient to refer to papers by physicist, written say during the twelve years following the first publication of Maxwell's treatise, to see that there may be much difference of opinion as to what his theory is.
It may be, and has been, differently interpreted by different men, which is a sign that it is not set forth in a perfectly clear and unmistakeable form. There are many obscurities and some inconsistencies.
Speaking for myself, it was only by changing its form of presentation that I was able to see it clearly, and so as to avoid the inconsistencies. Now there is no finality in a growing science. It is, therefore, impossible to adhere strictly to Maxwell's theory as he gave it to the world, if only on account of its inconvenient form.
But it is clearly not admissible to make arbitrary changes in it and still call it his. He might have repudiated them utterly. But if we have good reason to believe that the theory as stated in his treatise does require modification to make it self-consistent, and to believe that he would have admitted the necessity of the change when pointed out to him, then I think the resulting modified theory may well be called Maxwell's."
The first approximation to the answer is to say, There is Maxwell's book as he wrote it; there is his text, and there are his equations: together they make his theory. But when we come to examine it closely, we find that this answer is unsatisfactory.
To begin with, it is sufficient to refer to papers by physicist, written say during the twelve years following the first publication of Maxwell's treatise, to see that there may be much difference of opinion as to what his theory is.
It may be, and has been, differently interpreted by different men, which is a sign that it is not set forth in a perfectly clear and unmistakeable form. There are many obscurities and some inconsistencies.
Speaking for myself, it was only by changing its form of presentation that I was able to see it clearly, and so as to avoid the inconsistencies. Now there is no finality in a growing science. It is, therefore, impossible to adhere strictly to Maxwell's theory as he gave it to the world, if only on account of its inconvenient form.
But it is clearly not admissible to make arbitrary changes in it and still call it his. He might have repudiated them utterly. But if we have good reason to believe that the theory as stated in his treatise does require modification to make it self-consistent, and to believe that he would have admitted the necessity of the change when pointed out to him, then I think the resulting modified theory may well be called Maxwell's."
[2] "On Compressional Electric or Magnetic Waves", O. Heaviside E.M.T. Vol. 2, Appendix D pgs. 493-506
[3] "On the Metaphysical Nature of the Propagation of the Potentials" O. Heaviside,Phil. Mag. 1889, Series 5, Vol. 27, No. 164, pgs 47-50
[4b] "On An Expression of the Electromagnetic Field Due to Electrons by Means of Two Scalar Potential Functions", E.T. Whittaker London Mathematical Society, 1904, Series 2, Vol. 1, pgs 367-372
[5] Helmholtz's paper is hard to come by, however, in reference [2] Heaviside brings it up and gives some commentary.
Regards,
Garrett M
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